Solve for $x$ : $ 8|x + 9| - 3 = 3|x + 9| + 4 $
Subtract $ {3|x + 9|} $ from both sides: $ \begin{eqnarray} 8|x + 9| - 3 &=& 3|x + 9| + 4 \\ \\ { - 3|x + 9|} && { - 3|x + 9|} \\ \\ 5|x + 9| - 3 &=& 4 \end{eqnarray} $ Add ${3}$ to both sides: $ \begin{eqnarray} 5|x + 9| - 3 &=& 4 \\ \\ { + 3} &=& { + 3} \\ \\ 5|x + 9| &=& 7 \end{eqnarray} $ Divide both sides by ${5}$ $ \dfrac{5|x + 9|} {{5}} = \dfrac{7} {{5}} $ Simplify: $ |x + 9| = \dfrac{7}{5}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 9 = -\dfrac{7}{5} $ or $ x + 9 = \dfrac{7}{5} $ Solve for the solution where $x + 9$ is negative: $ x + 9 = -\dfrac{7}{5} $ Subtract ${9}$ from both sides: $ \begin{eqnarray} x + 9 &=& -\dfrac{7}{5} \\ \\ {- 9} && {- 9} \\ \\ x &=& -\dfrac{7}{5} - 9 \end{eqnarray} $ Change the ${ - 9}$ to an equivalent fraction with a denominator of $5$ $ x = - \dfrac{7}{5} {- \dfrac{45}{5}} $ $ x = -\dfrac{52}{5} $ Then calculate the solution where $x + 9$ is positive: $ x + 9 = \dfrac{7}{5} $ Subtract ${9}$ from both sides: $ \begin{eqnarray} x + 9 &=& \dfrac{7}{5} \\ \\ {- 9} && {- 9} \\ \\ x &=& \dfrac{7}{5} - 9 \end{eqnarray} $ Change the ${ - 9}$ to an equivalent fraction with a denominator of $5$ $ x = \dfrac{7}{5} {- \dfrac{45}{5}} $ $ x = -\dfrac{38}{5} $ Thus, the correct answer is $x = -\dfrac{52}{5} $ or $x = -\dfrac{38}{5} $.